Geodesic
Approximative compactness of the algebraic sum of sets
Matematičeskie zametki, Tome 23 (1978) no. 1, pp. 55-60

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Let $X$ be a group with an invariant metric, $A$ and $B$ nonempty subsets of $X$ with $B$ compact. It is proved that if $A$ is an existence set [1] (approximatively compact [2]) then $A+B$ and $B+A$ are existence sets (approximatively compact). An example is given of a one-dimensional linear metric space (with an invariant metric) in which there exist an approximatively compact set $A$ and an element $v$ such that $A+v$ is not an existence set. 
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     author = {A. I. Vasil'ev},
     title = {Approximative compactness of the algebraic sum of sets},
     journal = {Matemati\v{c}eskie zametki},
     pages = {55--60},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_1_a5/}
}
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A. I. Vasil'ev. Approximative compactness of the algebraic sum of sets. Matematičeskie zametki, Tome 23 (1978) no. 1, pp. 55-60. http://geodesic.mathdoc.fr/item/MZM_1978_23_1_a5/